Strong underrelaxation in Kaczmarz's method for inconsistent systems

Yair Censor; Paul P.B. Eggermont; Dan Gordon

doi:10.1007/BF01396307

Strong underrelaxation in Kaczmarz's method for inconsistent systems

Yair Censor, Paul P.B. Eggermont, Dan Gordon

Department of Computer Science

Research output: Contribution to journal › Article › peer-review

Abstract

We investigate the behavior of Kaczmarz's method with relaxation for inconsistent systems. We show that when the relaxation parameter goes to zero, the limits of the cyclic subsequences generated by the method approach a weighted least squares solution of the system. This point minimizes the sum of the squares of the Euclidean distances to the hyperplanes of the system. If the starting point is chosen properly, then the limits approach the minimum norm weighted least squares solution. The proof is given for a block-Kaczmarz method.

Original language	English
Pages (from-to)	83-92
Number of pages	10
Journal	Numerische Mathematik
Volume	41
Issue number	1
DOIs	https://doi.org/10.1007/BF01396307
State	Published - Feb 1983

Keywords

Subject Classifications: AMS (MOS): 65F10, CR: 5.14

ASJC Scopus subject areas

Computational Mathematics
Applied Mathematics

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10.1007/BF01396307

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abstract = "We investigate the behavior of Kaczmarz's method with relaxation for inconsistent systems. We show that when the relaxation parameter goes to zero, the limits of the cyclic subsequences generated by the method approach a weighted least squares solution of the system. This point minimizes the sum of the squares of the Euclidean distances to the hyperplanes of the system. If the starting point is chosen properly, then the limits approach the minimum norm weighted least squares solution. The proof is given for a block-Kaczmarz method.",

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Strong underrelaxation in Kaczmarz's method for inconsistent systems

Abstract

Keywords

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